The present invention relates to apparatus and method for improved efficiency of adaptations of finite element meshes for numerical solutions of partial differential equations.
The finite-element method is the most effective of currently available numerical techniques for solving various problems arising from mathematical physics and engineering. In particular, it is the most widely used of numerical techniques for solving problems described by partial differential equations (PDEs).
Time-dependent PDEs arise when modeling numerous phenomena in science and engineering, and tend to be divided into two categories: hyperbolic and parabolic. The hyperbolic PDE is used for transient and harmonic wave propagation in acoustics and electromagnetics, and for transverse motions of membranes. The basic prototype of the hyperbolic PDE is the family of wave equations. The parabolic PDE is used for unsteady heat transfer in solids, flow in porous media and diffusion problems. The basic prototype parabolic PDE is the family of heat equations.
The concept behind the finite-element method is to reduce a continuous physical problem with infinitely many unknown field values to a finite number of unknowns by discretizing the solution region into elements. Then, the values of the field at any point can be approximated by interpolation functions within every element in terms of the field values at specified points called nodes. Nodes are located at the element vertices where adjacent elements are connected. The approximation of the solution at each element should be consistent with neighboring elements.
Several approaches can be used to transform the continuous physical formulation of the problem to its finite-element discrete analogue. For PDEs, the most popular method of their finite element formulation is that known as the Galerkin method as discussed in O. Axelsson and V. A. Barker. Finite Element Solution of Boundary Value Problems, Academic Press, Inc., London, 1984, the contents of which are hereby incorporated by reference.
In time-dependent problems, e.g. hyperbolic equations, areas of interest, i.e. areas with high gradient or with turning points or rapid changes, are propagated through the domain, that is they tend to move over time across the solution domain. Therefore, the mesh choice is preferably dynamic and follows the propagation of the areas of interest with time. For example, when the solution of hyperbolic problems involves a shock wave propagating through the mesh the location of the shock vicinity keeps changing in time. Thus, one wants to have the mesh more refined around the area of the shock vicinity and less refined elsewhere. Another example is the problem of fluid flow in a cavity, where flow cells are generated and undergo continuous changes in their shapes and size as time proceeds. Thus, the mesh adaptation itself is a crucial part of the efficient computation of the numerical method. In order to achieve an optimal mesh, that is one in which the solution error is low relative to the number of nodes in the mesh, the mesh choice needs to be dynamic and must vary with time.
Current systems use indicators such as gradients from the solution at a present stage to identify where the mesh should be modified, that is to say where it should be refined and where it can be made coarser, at the next time stage. However, such a system suffers from the obvious defect that it operates one step behind. In other words, if the areas of interest are propagated, then the current systems leave refining a step behind the most interesting phenomena.
There is thus a widely recognized need for, and it would be highly advantageous to have, a device and method for optimizing finite mesh based numerical solutions which is devoid of the above limitations.